Question

a parabola, with its vertex at the origin, has a directrix at ( y = 3 ).
which statements about the parabola are true? select two options.
( square ) the focus is located at ( (0,-3) ).
( square ) the parabola opens to the left.
( square ) the ( p ) value can be determined by computing ( 4(3) ).
( square ) the parabola can be represented by the equation ( x^{2}=-12y ).
( square ) the parabola can be represented by the equation ( y^{2}=12x ).

Explanation:

Step1: Recall the properties of a parabola

For a parabola with vertex at the origin \((0,0)\), if the directrix is \(y = k\), the standard form is \(x^{2}=4p(y - h)\) (where \((h,k)\) is the vertex). Here \(h = 0,k = 0\) and directrix \(y=3\). The formula for the directrix of \(x^{2}=4py\) is \(y=-p\). Since \(y = 3\) is the directrix, then \(p=- 3\).

Step2: Analyze each option

• Option 1:

The focus of \(x^{2}=4py\) is \((0,p)\). Since \(p=-3\), the focus is \((0,-3)\).

• Option 2:

Since the directrix is \(y = 3\) and the vertex is at the origin, the parabola opens downwards (not to the left). The general form for a parabola opening left is \(y^{2}=-4px\) (\(p>0\)).

• Option 3:

The formula for the directrix is \(y=-p\). If \(y = 3\) (directrix), then \(p=-3\), not \(4\times3\).

• Option 4:

Substitute \(p=-3\) into \(x^{2}=4py\). We get \(x^{2}=4\times(- 3)y=-12y\).

• Option 5:

The form \(y^{2}=12x\) is a parabola that opens to the right (\(4p = 12\Rightarrow p = 3\), directrix \(x=-3\)) which is not our case.

Answer:

The focus is located at \((0,-3)\); The parabola can be represented by the equation \(x^{2}=-12y\) (i.e., the first and the fourth options).